[seqfan] Re: defn of A032452 transients in modified Poulet
Neil Sloane
njasloane at gmail.com
Sun May 10 00:27:38 CEST 2020
Thanks to Amiram Eldar who first saw (almost) what was going on with
A032452, and John Machacek who found the precise explanation.
I have edited the original sequence, preserving the terms and modifying the
definition to match the data (that is the preferred way to change an
existing sequence - the data takes precedence). Amiram created another
version, A334725, which I have edited and approved, and I created another
new sequence A334523 for what I think is a better version of the original
sequence (starting each row with n).
By the way, both Amiram and John (in his program) seem to assume that 1
will always be reached. This is an open question, I believe, so I
changed all the definitions to add an escape clause to say what the
sequence does if some value of n never reaches 1. There could be a loop
not containing 1, or the sequence could blow up, though that seems very
unlikely.
So I think this wraps up A032452.
However, after reading the Alaoglu-Erdos paper, one sees that several
similar or related sequences could be created
If we iterate phi, sigma, phi, how many steps does it take to read 1, for
example? (with -1 if that doesn't happen)
And look at (a) at the end of the paper. Which n's don't go to infinity?
This could be a sequence (stopping of course at the first value for which
there is no proof of divergence)
Best regards
Neil
Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com
On Sat, May 9, 2020 at 12:24 PM Ami Eldar <amiram.eldar at gmail.com> wrote:
> Hello again,
>
> I have just seen that John Machacek found the exact formulation that gives
> data without errors.
> It is (phi, sigma, phi), without the initial term n.
>
> So I have changed my draft
> https://oeis.org/history/view?seq=A032452&v=18
> to the current
> https://oeis.org/draft/A032452
> without the correction of the data.
>
> Please see which version you prefer.
> I will then correct A334725 accordingly, if necessary.
>
> Best,
> Amiram
>
> On Sat, May 9, 2020 at 10:31 AM Ami Eldar <amiram.eldar at gmail.com> wrote:
>
>>
>> Hello Neil,
>>
>> I have edited A032452 and created a new sequence with the order (phi,
>> phi, sigma).
>>
>> Both are now in the edit stack. Please see
>> https://oeis.org/draft/A032452
>> and
>> https://oeis.org/draft/A334725
>>
>> Best,
>> Amiram
>>
>> ---------- Forwarded message ---------
>> From: John Machacek <jmachacek.math at gmail.com>
>> Date: Sat, May 9, 2020 at 7:26 AM
>> Subject: [seqfan] Re: defn of A032452 transients in modified Poulet
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>>
>>
>> Hi,
>>
>> I believe it is iterates of phi, sigma, phi, phi, sigma, phi, phi, sigma,
>> phi,... until reaching 1 for each positive integer n=1,2,3,4.... When
>> going
>> to the next integer we "skip" that integer and start with phi(n).
>>
>> Here is A032452 split in this way.
>>
>> 1,
>> 1,
>> 2, 3, 2, 1,
>> 2, 3, 2, 1,
>> 4, 7, 6, 2, 3, 2, 1,
>> 2, 3, 2, 1,
>> 6, 12, 4, 2, 3, 2, 1,
>> 4, 7, 6, 2, 3, 2, 1,
>> 6, 12, 4, 2, 3,...
>>
>> Notice the first entry in each row is phi(n). The SageMath code below
>> reproduces A032452.
>>
>> Best,
>> John
>>
>> N = 10
>>
>> n = 1
>>
>> seq = []
>>
>> while n < N:
>>
>> a = euler_phi(n)
>>
>> seq.append(a)
>>
>> i = 0
>>
>> while a != 1:
>>
>> if i%3 == 0:
>>
>> a = sigma(a)
>>
>> else:
>>
>> a = euler_phi(a)
>>
>> seq.append(a)
>>
>> i += 1
>>
>> n += 1
>>
>> On Fri, May 8, 2020 at 9:53 PM Neil Sloane <njasloane at gmail.com> wrote:
>>
>> > All those sequences have a (free) link to the paper - see under the
>> LINKS
>> > section!
>> >
>> > Best regards
>> > Neil
>> >
>> > Neil J. A. Sloane, President, OEIS Foundation.
>> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway,
>> NJ.
>> > Phone: 732 828 6098; home page: http://NeilSloane.com
>> > Email: njasloane at gmail.com
>> >
>> >
>> >
>> > On Fri, May 8, 2020 at 7:51 PM Tom Duff <td at pixar.com> wrote:
>> >
>> > > Got it here: http://iq0.com/alaoglu-erdos-conjecture.pdf
>> > > It's only two pages.
>> > >
>> > > On Fri, May 8, 2020 at 4:28 PM Marc LeBrun <mlb at well.com> wrote:
>> > > >
>> > > > >=Neil Sloane
>> > > > > I thought some more about this problem, without success.
>> > > >
>> > > > Perhaps if someone has access to that Alaoglu & Erdös reference it
>> > might
>> > > help?
>> > > >
>> > > >
>> > > > --
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>> >
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>>
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>
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